where \(\bar{X}_{T(R)} = \frac{1}{n} \sum _{i=1}^n X_{T(R)}^{(i)}\) . Finally, we use the Hoeffding 1963 theorem [1]} to show that
\(\hat{q}_\alpha = \min \lbrace  \bar{X}_T, \bar{X}_R\rbrace  - \sqrt{\frac{16 \log (1/\alpha )}{2 n}}\) 
